Stackel Spaces Research Articles

Classification of the non-null electrovacuum solution of Einstein–Maxwell equations with three-parameter abelian group of motions

The classification of the Stackel spaces of the electrovacuum of the type (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. In the case of a non-zero cosmological term, the metrics and potentials contain solutions of a nonlinear ordinary differential equation of the second order. When the cosmological term equals zero, the metrics and the components of the electromagnetic field tensor are expressed through elementary functions. Thus the classification of the electrovacuum stackel spaces of all types is completed and complete list of these spaces is constructed.

Separation of variables in Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle in the stackel spaces of type (1.1)

All equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces provided that Hamilton–Jacobi equation and Klein–Gordon–Fock equation for a charged test particle can be integrated by the method of complete separation of variables are found. The separation is carried out using the complete sets of mutually commuting integrals of motion of type (1.1) whereby in a privileged coordinate system, the given equations turn into parabolic type equations. Hence, these metrics can be used as models for describing plane gravitational waves.

Separation of variables in Hamilton–Jacobi equation for a charged test particle in the Stackel spaces of type (2.1)

We can find all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of variables in the Hamilton–Jacobi equation. Separation is carried out using the complete sets of mutually-commuting integrals of motion of type (2.1), whereby in a privileged coordinate system the Hamilton–Jacobi equation turns into a parabolic type equation.

Spatially Homogeneous Conformally Stäckel Spaces of Type (3.1)

In this work, we have obtained all spatially homogeneous space-time models related to the intersection of the set of Stackel spaces of type (2.1) and the set of conformally Stackel spaces of type (3.1). These models allow a complete separation of variables both in the Hamilton-Jacobi equation for massive test particles and in the eikonal equation for radiation. The models obtained in this work relate to wave-like space-time models. For the found models, solutions of the Einstein equations with the cosmological constant and radiation are obtained. For the obtained solutions, the eikonal equation and the equations of motion of massive test particles in the Hamilton-Jacobi form have been integrated.

Пространственно-однородные конформно-штеккелевы пространства типа (3.1)

Пространственно-однородные конформно-штеккелевы пространства типа (3.1)

Пространственно-однородные модели штеккелевых пространств типа (2.1)

Пространственно-однородные модели штеккелевых пространств типа (2.1)

The Vaidya problem in conformally flat Stäckel spaces of type (1.1)

The problem of classification of conformally flat radiation spaces admitting full separation of variables in the Hamilton–Jacobi equation is considered. The Vaidya problem for conformally flat Stackel spaces of type (1.1) has been solved; the metrics and the radiation have been found in explicit form.

Integration of the dirac equation, which does not presume complete separation of variables, in Stäckel spaces

The method of noncommutative integration of linear differential equations is used to construct an exact solution of the Dirac equation, which does not presume complete separation of variables, in Stackel spaces. The Dirac equation in an external electromagnetic field is integrated by this method, using one example. The Stackel space under consideration does not enable one to solve this equation exactly within the framework of the theory of separation of variables.

Complete separation of variables in the free Hamilton-Jacobi equation

The theory of Stackel spaces is generalized to the case when the coordinate system in which there is complete separation of variables in the free Hamilton-Jacobi equation contains complex variables. Theorems which establish necessary and sufficient criteria of such spaces are proved.

A complete classification of St�ckel spaces of type (2.1) in the Brans-Dicke theory

In the paper the complete classification of vacuum stackel spaces of type (2.1) in the Brans-Dicke theory is completed.

St�ckel electrovacuum spaces that admit full separation of variables in a diagonalized Dirac equation. Type (2.1)

We list all electrovacuum Stackel spaces of the type (2.1) that admit diagonalization and full separation of variables in the Dirac equation.

Separation of variables in the Dirac equation in Stackel spaces. II. External gauge fields

For pt.I see ibid., vol.7, p.517, (1990). This is the second paper in a series of research started by Bagrov et al. The Abelian complete sets of matrix first-order differential symmetry operators for the Dirac equation are used for the separation of variables in the Dirac equation in Stackel spaces with external gauge fields. The permissible space metrics and gauge field potentials are not limited by field equations.

Separation of variables in the Dirac equation in Stackel spaces

The subspaces of Riemannian space of signature (+---) that admit separation of the Dirac equation have been found in the case of Riemannian space admitting the separation of the Hamilton-Jacobi equation. For the separation of variables in the Hamilton-Jacobi equation it is necessary for the complete set of Killing vectors and tensors to be of a special kind. Every complete set defines its own type of metric of Riemannian space which is called Stackel space. The Dirac equation does not permit the separation of variables in general cases of Stackel space. The main idea of the paper is in the construction, in Stackel space, of a complete set of another kind. This complete set consists of three matrix first-order differential symmetry operators of the Dirac equation. The operators are pairwise commuting and linearly independent. The complete set structure is in agreement with the structure of the Killing vectors and tensors of Stackel space. The separation of variables in the Dirac equation has been carried out in the explicit form in Stackel spaces which admit complete sets of symmetry operators. These operators have been used essentially in the process of separation that differs from Chandrasekhar method.

Stackel spaces of an electrovacuum with isotropic complete sets. Formulation of problem and basic relations

We consider the problem of the classification of the Stackel spaces of the electrovacuum with isotropic complete sets. The metrics of the spaces are represented in a form that is convenient for their investigation. We obtain necessary relations for the construction of the field equations.

Structure of the generalized momentum of a charged test particle and the inverse problem in the general theory of relativity

The work of Sedov [1] examined the navigational problem which involves finding the metric tensor giθ of space of general relativity from the results of physical measurements. The problem of the approximate determination of giθ from the mutual deflection of test particles in gravitational fields was studied in [2], The problem of finding giθ from the second integrals of the geodesic equations in gravitational fields was solved in [3]. The connection of the first integrals of the Hamilton-Jacobi equation of the test particles with the symmetry of StAckel spaces was investigated by Shapovalov [4, 5].